The Kannan–Lovász–Simonovits spectral-gap conjecture

Let $X$ be a centered log-concave random vector in $\mathbb R^n$, let $\lambda_X^2$ be the largest eigenvalue of its covariance matrix, and let $g:\mathbb R^n\to\mathbb R$ be locally Lipschitz. The Kannan–Lovász–Simonovits conjecture. There exists an absolute constant $C$ such that

$$\operatorname{Var}g(X)\leq C\lambda_X^2\mathbb E|\nabla g(X)|^2.$$

This is a spectral-gap conjecture for log-concave probability measures and implies the variance conjecture by taking $X$ isotropic and $g(X)=|X|^2$. Its status is not resolved in the supplied text.